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Non-transitive points and porosity

100%

Subrahmonian Moothathu T.

Colloquium Mathematicum

2013

tom 133

nr 1

99-114

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show that for a family of piecewise monotonic transitive interval maps, the set of non-transitive points is σ-polynomially porous. We indicate how similar methods can be used to give sufficient conditions for the set of non-recurrent points and the set of distal pairs of a dynamical system to be very small.

Diagonal points having dense orbit

100%

Subrahmonian Moothathu T.

Colloquium Mathematicum

2010

tom 120

nr 1

127-138

Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f × f² × ⋯ × f^{m}: X^{m} → X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f × f² × ⋯ × f^{m}$. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).

Ograniczanie wyników

2 Colloquium Mathematicum

2 Subrahmonian Moothathu T.

1 2013

1 2010

Wyniki wyszukiwania

Non-transitive points and porosity

Diagonal points having dense orbit